Vernier measurement techniques can improve the resolution of an instrument that makes a coarse measurement with a main scale. Generally, a Vernier scale has an integer number of divisions, D, that span the same length or arc as D−1 divisions of the main scale. The Vernier scale can be read by locating the best-aligned division mark on the Vernier scale that aligns with any mark on the main scale. The best-aligned Vernier scale mark augments the main scale measurement with a high-resolution length or arc increment.
The Vernier scale can increase the resolution of the instrument because the numbers of divisions of the two scales are coprime integers. For example, D is coprime to D−1 for any integer D greater than one. The resolution of an instrument equipped with a Vernier scale that has D divisions that span the same length or arc as D−1 main scale divisions can be D times better than one division of the main scale. The number of main scale or Vernier scale divisions required to achieve a high-resolution may be large.
Vernier techniques may be applied to time difference measurement problems by using delay elements to determine time divisions. For example, Vernier techniques may be used to measure the difference in the time of arrival or phase of two electronic signals. The delay elements may have a minimum manufacturable delay value that can preclude direct high-resolution measurements of time differences. The number of delay elements needed to achieve high-resolution time difference measurements may be large. The uncertainty in the measured time difference can increase with the number of delay elements, thus affecting devices that are sensitive to time or phase measurement errors.